MACLAURIN’S SERIES 71A
- Use Maclaurin’s series to determine the
expansion of (3+ 2 t)^4.
[
81 + 216 t+ 216 t^2 + 96 t^3 + 16 t^4
]8.5 Numerical integration using
Maclaurin’s series
The value of many integrals cannot be determined
using the various analytical methods. In Chapter
45, the trapezoidal, mid-ordinate and Simpson’s
rules are used to numerically evaluate such inte-
grals. Another method of finding the approximate
value of a definite integral is to express the func-
tion as a power series using Maclaurin’s series, and
then integrating each algebraic term in turn. This is
demonstrated in the following worked problems.
Problem 11. Evaluate∫ 0. 4
0. 1 2esinθdθ, correct
to 3 significant figures.A power series for esinθ is firstly obtained using
Maclaurin’s series.
f(θ)=esinθ f(0)=esin 0=e^0 = 1f′(θ)=cosθesinθ f′(0)=cos 0 esin 0=(1)e^0 = 1f′′(θ)=(cosθ)(cosθesinθ)+(esinθ)(−sinθ),
by the product rule,=esinθ(cos^2 θ−sinθ);f′′(0)=e^0 (cos^20 −sin 0)= 1f′′′(θ)=(esinθ)[(2 cosθ(−sinθ)−cosθ)]+(cos^2 θ−sinθ)(cosθesinθ)=esinθcosθ[−2 sinθ− 1 +cos^2 θ−sinθ]f′′′(0)=e^0 cos 0[(0− 1 + 1 −0)]= 0
Hence from equation (5):
esinθ=f(0)+θf′(0)+
θ^2
2!f′′(0)+θ^3
3!f′′′(0)+···= 1 +θ+θ^2
2+ 0Thus∫ 0. 40. 12esinθdθ=∫ 0. 40. 12(
1 +θ+θ^2
2)
dθ=∫ 0. 40. 1(2+ 2 θ+θ^2 )dθ=[
2 θ+2 θ^2
2+θ^3
3] 0. 40. 1=(
0. 8 +(0.4)^2 +(0.4)^3
3)−(
0. 2 +(0.1)^2 +(0.1)^3
3)= 0. 98133 − 0. 21033
= 0. 771 , correct to 3 significant figuresProblem 12. Evaluate∫ 10sinθ
θdθ usingMaclaurin’s series, correct to 3 significant
figures.Let f(θ)=sinθ f(0)= 0
f′(θ)=cosθ f′(0)= 1
f′′(θ)=−sinθ f′′(0)= 0
f′′′(θ)=−cosθ f′′′(0)=− 1fiv(θ)=sinθ fiv(0)= 0
fv(θ)=cosθ fv(0)= 1Hence from equation (5):sinθ=f(0)+θf′(0)+θ^2
2!f′′(0)+θ^3
3!f′′′(0)+θ^4
4!fiv(0)+θ^5
5!fv(0)+···= 0 +θ(1)+θ^2
2!(0)+θ^3
3!(−1)+θ^4
4!(0)+θ^5
5!(1)+···i.e. sinθ=θ−θ^3
3!+θ^5
5!−···Hence∫ 10sinθ
θdθ=∫ 10(θ−θ^3
3!+θ^5
5!−θ^7
7!+···)θdθ=∫ 10(
1 −θ^2
6+θ^4
120−θ^6
5040+···)
dθ